Solve for $x$ : $ 2|x - 2| - 4 = 1|x - 2| + 8 $
Explanation: Subtract $ {1|x - 2|} $ from both sides: $ \begin{eqnarray} 2|x - 2| - 4 &=& 1|x - 2| + 8 \\ \\ { - 1|x - 2|} && { - 1|x - 2|} \\ \\ 1|x - 2| - 4 &=& 8 \end{eqnarray} $ Add ${4}$ to both sides: $ \begin{eqnarray} 1|x - 2| - 4 &=& 8 \\ \\ { + 4} &=& { + 4} \\ \\ 1|x - 2| &=& 12 \end{eqnarray} $ Simplify: $ |x - 2| = 12$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 2 = -12 $ or $ x - 2 = 12 $ Solve for the solution where $x - 2$ is negative: $ x - 2 = -12 $ Add ${2}$ to both sides: $ \begin{eqnarray} x - 2 &=& -12 \\ \\ {+ 2} && {+ 2} \\ \\ x &=& -12 + 2 \end{eqnarray} $ $ x = -10 $ Then calculate the solution where $x - 2$ is positive: $ x - 2 = 12 $ Add ${2}$ to both sides: $ \begin{eqnarray} x - 2 &=& 12 \\ \\ {+ 2} && {+ 2} \\ \\ x &=& 12 + 2 \end{eqnarray} $ $ x = 14 $ Thus, the correct answer is $x = -10 $ or $x = 14 $.